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Berlekamp–Massey algorithm

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Title: Berlekamp–Massey algorithm  
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Subject: Reed–Solomon error correction, Berlekamp–Welch algorithm, Rank error-correcting code, Elwyn Berlekamp, Cryptanalytic algorithms
Collection: Cryptanalytic Algorithms, Error Detection and Correction
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Berlekamp–Massey algorithm

The Berlekamp–Massey algorithm is an algorithm that will find the shortest linear feedback shift register (LFSR) for a given binary output sequence. The algorithm will also find the minimal polynomial of a linearly recurrent sequence in an arbitrary field. The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse.[1] Reeds and Sloane offer an extension to handle a ring.[2]

Elwyn Berlekamp invented an algorithm for decoding Bose–Chaudhuri–Hocquenghem (BCH) codes.[3][4] James Massey recognized its application to linear feedback shift registers and simplified the algorithm.[5][6] Massey termed the algorithm the LFSR Synthesis Algorithm (Berlekamp Iterative Algorithm),[7] but it is now known as the Berlekamp–Massey algorithm.

Contents

  • Description of algorithm 1
  • Code sample 2
  • The algorithm for the binary field 3
  • Code sample for the binary field in Java 4
  • See also 5
  • References 6
  • External links 7

Description of algorithm

The Berlekamp–Massey algorithm is an alternate method to solve the set of linear equations described in Reed–Solomon Peterson decoder, which can be summarized as:

S_{i + \nu} + \Lambda_1 S_{i+\nu-1} + \cdots + \Lambda_{\nu-1} S_{i+1} + \Lambda_{\nu} S_i = 0.

In the code examples below, C(x) is a potential instance of Λ(x). The error locator polynomial C(x) for L errors is defined as:

C(x) = C_{L} \ x^{L} + C_{L-1} \ x^{L-1} + \cdots + C_2 \ x^2 + C_1 \ x + 1

or reversed:

C(x) = 1 + C_1 \ x + C_2 \ x^2 + \cdots + C_{L-1} \ x^{L-1} + C_{L} \ x^{L}.

The goal of the algorithm is to determine the minimal degree L and C(x) which results in:

S_{n} + C_1 \ S_{n-1} + \cdots + C_L \ S_{n-L} = 0

for all syndromes, n = L to (N-1).

Algorithm: C(x) is initialized to 1, L is the current number of assumed errors, and initialized to zero. N is the total number of syndromes. n is used as the main iterator and to index the syndromes from 0 to (N-1). B(x) is a copy of the last C(x) since L was updated and initialized to 1. b is a copy of the last discrepancy d (explained below) since L was updated and initialized to 1. m is the number of iterations since L, B(x), and b were updated and initialized to 1.

Each iteration of the algorithm calculates a discrepancy d. At iteration k this would be:

d = S_{k} + C_1 \ S_{k-1} + \cdots + C_L \ S_{k-L}.

If d is zero, the algorithm assumes that C(x) and L are correct for the moment, increments m, and continues.

If d is not zero, the algorithm adjusts C(x) so that a recalculation of d would be zero:

C(x) = C(x) \ - \ (d / b) \ x^m \ B(x).

The xm term shifts B(x) so it follows the syndromes corresponding to 'b'. If the previous update of L occurred on iteration j, then m = k - j, and a recalculated discrepancy would be:

d = S_{k} + C_1 \ S_{k-1} + \cdots - (d/b) (S_{j} + B_1 \ S_{j-1} + \cdots ).

This would change a recalculated discrepancy to:

d = d - (d/b)b = d - d = 0. \

The algorithm also needs to increase L (number of errors) as needed. If L equals the actual number of errors, then during the iteration process, the discrepancies will become zero before n becomes greater than or equal to (2 L). Otherwise L is updated and algorithm will update B(x), b, increase L, and reset m = 1. The formula L = (n + 1 - L) limits L to the number of available syndromes used to calculate discrepancies, and also handles the case where L increases by more than 1.

Code sample

The algorithm from Massey (1969, p. 124).

  polynomial(field K) s(x) = ... /* coeffs are s_j; output sequence as N-1 degree polynomial) */
  /* connection polynomial */
  polynomial(field K) C(x) = 1;  /* coeffs are c_j */
  polynomial(field K) B(x) = 1;
  int L = 0;
  int m = 1;
  field K b = 1;
  int n;
  for (n = 0; n < N; n++)
    {
      /* calculate discrepancy */
      field K d = s_n + \Sigma_{i=1}^L c_i * s_{n-i};
      if (d == 0)
        {
          /* annihilation continues */
          m = m + 1;
        }
      else if (2 * L <= n)
        {
          /* temporary copy of C(x) */
          polynomial(field K) T(x) = C(x);
          C(x) = C(x) - d b^{-1} x^m B(x);
          L = n + 1 - L;
          B(x) = T(x);
          b = d;
          m = 1;
        }
      else
        {
          C(x) = C(x) - d b^{-1} x^m B(x);
          m = m + 1;
        }
    }
  return L;

The algorithm for the binary field

The following is the Berlekamp–Massey algorithm specialized for the typical binary finite field F2 and GF(2). The field elements are 0 and 1. The field operations + and − are identical and become the exclusive or operation, XOR. The multiplication operator * becomes the logical AND operation. The division operator reduces to the identity operation (i.e., field division is only defined for dividing by 1, and x/1 = x).

  1. Let s_0, s_1, s_2 \cdots s_{n-1} be the bits of the stream.
  2. Initialise two arrays b and c each of length n to be zeroes, except b_0 \leftarrow 1, c_0 \leftarrow 1
  3. assign L \leftarrow 0, m \leftarrow -1.
  4. For N = 0 step 1 while N < n :
    • Let d be s_N + c_1s_{N-1} + c_2s_{N-2} + \cdots + c_Ls_{N-L}.
    • if d = 0, then c is already a polynomial which annihilates the portion of the stream from N-L to N.
    • else:
      • Let t be a copy of c.
      • Set c_{N-m} \leftarrow c_{N-m} \oplus b_0, c_{N-m+1} \leftarrow c_{N-m+1} \oplus b_1, \dots up to c_{n-1} \leftarrow c_{n-1} \oplus b_{n-N+m-1} (where \oplus is the Exclusive or operator).
      • If L \le \frac{N}{2}, set L \leftarrow N+1-L, set m \leftarrow N, and let b \leftarrow t; otherwise leave L, m and b alone.

At the end of the algorithm, L is the length of the minimal LFSR for the stream, and we have c_Ls_a + c_{L-1}s_{a+1} + c_{L-2}s_{a+2} + \cdots = 0 for all a.

Code sample for the binary field in Java

The following code sample is for a binary field.

    public static int runTest(int[] array) {
        final int N = array.length;
        int[] b = new int[N];
        int[] c = new int[N];
        int[] t = new int[N];
        b[0] = 1;
        c[0] = 1;
        int l = 0;
        int m = -1;
        for (int n = 0; n < N; n++) {
            int d = 0;
            for (int i = 0; i <= l; i++) {
                d ^= c[i] * array[n - i];
            }
            if (d == 1) {
                System.arraycopy(c, 0, t, 0, N);
                int N_M = n - m;
                for (int j = 0; j < N - N_M; j++) {
                    c[N_M + j] ^= b[j];
                }
                if (l <= n / 2) {
                    l = n + 1 - l;
                    m = n;
                    System.arraycopy(t, 0, b, 0, N);
                }
            }
        }
        return l;
    }

See also

References

  1. ^ Reeds & Sloane 1985, p. 2
  2. ^ Reeds, J. A.;  
  3. ^  
  4. ^  . Previous publisher McGraw-Hill, New York, NY.
  5. ^  
  6. ^ Ben Atti, Nadia; Diaz-Toca, Gema M.; Lombardi, Henri, The Berlekamp–Massey Algorithm revisited,  
  7. ^ Massey 1969, p. 124

External links

  • Hazewinkel, Michiel, ed. (2001), "Berlekamp-Massey algorithm",  
  • Berlekamp–Massey algorithm at PlanetMath.
  • Weisstein, Eric W., "Berlekamp–Massey Algorithm", MathWorld.
  • GF(2) implementation in Mathematica
  • (German) Applet Berlekamp–Massey algorithm
  • Online GF(2) Berlekamp-Massey calculator
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